Mathematical Sciences: Algebraic Structure of Quantum Groups

该奖项支持对非标准量子群的代数结构分析的研究。该项目的具体目标是：(1)沿着Hodges和Levasseur先前关于标准和多参数情况的工作，详细分析Cremmer-Gervais量子群的代数结构；(2)研究了相关泊松群和李双代数的结构，并与相关量子群的结构进行了比较；(3)构造进一步的非标准量子群族。本研究属于环理论的一般范畴。环是一个代数对象，在它上面定义了加法和乘法。虽然加法运算满足交换律，但乘法运算不需要满足交换律。一个环的例子，其乘法不可交换是整数上的nxn个矩阵的集合。非交换环的研究由于其对其他数学和物理分支的重要意义而成为代数的重要组成部分。

This award supports research on the analysis of the algebraic structure of non-standard quantum groups. The specific aims of the project are: (1) To analyze in detail the algebraic structure of the Cremmer-Gervais quantum groups along the lines of previous work of Hodges and Levasseur on the standard and multi-parameter cases; (2) To study the structure of the associated Poisson groups and Lie bialgebras and to compare this with the structure of the associated quantum group; (3) To construct further families of non-standard quantum groups. This research is in the general area of ring theory. A ring is an algebraic object having both an addition and a multiplication defined on it. Although the additive operation satisfies the commutative law, the multiplicative operation is not required to do so. An example of a ring for which multiplication in not commutative is the collection of nxn matrices over the integers. The study of noncommutative rings has become an important part of algebra because of its increasing significance to other branches of mathematics and physics.